Suggest is a tricky verb in English. We use it in lots of different grammatical structures, but there are some where we don't use it as well. For example, we might 'suggest doing something' but we don't say 'suggest to do' or 'suggest me'. These phrases are wrong.
Also when you want to make a suggestion, to sound natural, you might need to use a phrase without the verb suggest. Watch the video to see examples and avoid mistakes.